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Báo cáo toán học: " A Bijective Proof of Garsia’s q-Lagrange Inversion Theorem"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A Bijective Proof of Garsia’s q-Lagrange Inversion Theorem. | A Bijective Proof of Garsia s q-Lagrange Inversion Theorem Dan W. Singer Tiernan Communications 5751 Copley Drive San Diego CA 92111 dsinger@tiernan.com Submitted March 4 1997 Accepted April 25 1998 Abstract A q-Lagrange inversion theorem due to A. M. Garsia is proved by means of two sign-reversing weight-preserving involutions on Catalan trees. 1 Introduction Let F u be a formal power series with F 0 0 F0 0 0 delta series . Then F u has an inverse f u which satishes 1 p u k gn f u n uk n k and 1 f u k un F u n uk n k for all k 1 where un means extract the coefficient of un. The coefficients of f u n may be expressed in terms of the coefficients of F u by means of the Lagrange inversion formula f u n uk unF 0 u F u k 1 u 1 AMS Subject Classification 05E99 primary 05A17 secondary Keywords q-Lagrange inversion Catalan trees THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1997 R26 2 The q-Lagrange inversion problem may be stated as follows given a delta series F u and a sequence of formal power series Fk u g which is a q-analogue of F u k find fk u such that 1 X Fk u lu fn u uk 1.1 n k and 1 X fk u un Fn u uk 1.2 n k for all k 1. If fk u satisfies equations 1.1 and 1.2 then fk u is a q-analogue of f u k for each k where f u F-1 u . We say that Fk u g and fk u are inverse sequences. There are several solutions to the q-Lagrange inversion problem appearing in the literature see for example Andrews 2 Garsia 7 Garsia and Remmel 9 Gessel 10 Gessel and Stanton 11 12 Hofbauer 13 Krattenthaler 15 Singer 17 18 . Singer 17 proved an inversion theorem based on a generalization of Garsia s operator techniques which unifies and extends the q-Lagrange inversion theorems of Garsia 7 and Garsia-Remmel 9 . Garsia Gessel and Stanton and Singer have shown that Rogers-Ramanujan type identities may be derived by means of q-Lagrange inversion. Several authors have given quite distinct bijective proofs of q-series identities many of which may be interpreted as statements about partitions - see .